3.462 \(\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=291 \[ -\frac{171 a^2 \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{171 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{1024 d}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a \sin (c+d x)+a}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{28 d} \]

[Out]

(-171*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d) - (171*a^2*Cot[c + d*x])/(102
4*d*Sqrt[a + a*Sin[c + d*x]]) - (57*a^2*Cot[c + d*x]*Csc[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) + (199*a^2
*Cot[c + d*x]*Csc[c + d*x]^2)/(640*d*Sqrt[a + a*Sin[c + d*x]]) + (1237*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*
d*Sqrt[a + a*Sin[c + d*x]]) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c +
d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(28*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2
))/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 1.06083, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2881, 2762, 21, 2772, 2773, 206, 3044, 2975, 2980} \[ -\frac{171 a^2 \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{171 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{1024 d}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a \sin (c+d x)+a}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{28 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(-171*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d) - (171*a^2*Cot[c + d*x])/(102
4*d*Sqrt[a + a*Sin[c + d*x]]) - (57*a^2*Cot[c + d*x]*Csc[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) + (199*a^2
*Cot[c + d*x]*Csc[c + d*x]^2)/(640*d*Sqrt[a + a*Sin[c + d*x]]) + (1237*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*
d*Sqrt[a + a*Sin[c + d*x]]) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c +
d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(28*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2
))/(7*d)

Rule 2881

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Dist[1/d^4, Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^
n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&
!IGtQ[m, 0]

Rule 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 2772

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x]
+ Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 2773

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(-2*
b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3044

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n +
2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b
*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0
])

Rule 2975

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*S
in[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])
^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n +
1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d
, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n]
|| EqQ[c, 0])

Rule 2980

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n + 1)
*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^8(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac{\int \csc ^7(c+d x) \left (\frac{3 a}{2}-\frac{19}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{7 a}-\frac{1}{3} a \int \frac{\csc ^3(c+d x) \left (-\frac{11 a}{2}-\frac{11}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac{\int \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)} \left (-\frac{189 a^2}{4}-\frac{201}{4} a^2 \sin (c+d x)\right ) \, dx}{42 a}+\frac{1}{6} (11 a) \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac{1}{8} (11 a) \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{1}{560} (1237 a) \int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{11 a^2 \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac{1}{16} (11 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{1}{640} (1237 a) \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{11 a^2 \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a+a \sin (c+d x)}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac{1}{768} (1237 a) \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{\left (11 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}\\ &=-\frac{11 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}-\frac{11 a^2 \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a+a \sin (c+d x)}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a+a \sin (c+d x)}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac{(1237 a) \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{1024}\\ &=-\frac{11 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}-\frac{171 a^2 \cot (c+d x)}{1024 d \sqrt{a+a \sin (c+d x)}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a+a \sin (c+d x)}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a+a \sin (c+d x)}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac{(1237 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{2048}\\ &=-\frac{11 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}-\frac{171 a^2 \cot (c+d x)}{1024 d \sqrt{a+a \sin (c+d x)}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a+a \sin (c+d x)}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a+a \sin (c+d x)}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}+\frac{\left (1237 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{1024 d}\\ &=-\frac{171 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{1024 d}-\frac{171 a^2 \cot (c+d x)}{1024 d \sqrt{a+a \sin (c+d x)}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a+a \sin (c+d x)}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a+a \sin (c+d x)}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{28 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}\\ \end{align*}

Mathematica [A]  time = 4.76426, size = 522, normalized size = 1.79 \[ \frac{a \csc ^{22}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (306488 \sin \left (\frac{1}{2} (c+d x)\right )-177170 \sin \left (\frac{3}{2} (c+d x)\right )-6566 \sin \left (\frac{5}{2} (c+d x)\right )-219540 \sin \left (\frac{7}{2} (c+d x)\right )-33292 \sin \left (\frac{9}{2} (c+d x)\right )-3990 \sin \left (\frac{11}{2} (c+d x)\right )-11970 \sin \left (\frac{13}{2} (c+d x)\right )-306488 \cos \left (\frac{1}{2} (c+d x)\right )-177170 \cos \left (\frac{3}{2} (c+d x)\right )+6566 \cos \left (\frac{5}{2} (c+d x)\right )-219540 \cos \left (\frac{7}{2} (c+d x)\right )+33292 \cos \left (\frac{9}{2} (c+d x)\right )-3990 \cos \left (\frac{11}{2} (c+d x)\right )+11970 \cos \left (\frac{13}{2} (c+d x)\right )-209475 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+209475 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+125685 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-125685 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-41895 \sin (5 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+41895 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+5985 \sin (7 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-5985 \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{35840 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(a*Csc[(c + d*x)/2]^22*Sqrt[a*(1 + Sin[c + d*x])]*(-306488*Cos[(c + d*x)/2] - 177170*Cos[(3*(c + d*x))/2] + 65
66*Cos[(5*(c + d*x))/2] - 219540*Cos[(7*(c + d*x))/2] + 33292*Cos[(9*(c + d*x))/2] - 3990*Cos[(11*(c + d*x))/2
] + 11970*Cos[(13*(c + d*x))/2] + 306488*Sin[(c + d*x)/2] - 209475*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]
]*Sin[c + d*x] + 209475*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 177170*Sin[(3*(c + d*x))/2
] - 6566*Sin[(5*(c + d*x))/2] + 125685*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 125685*
Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 219540*Sin[(7*(c + d*x))/2] - 33292*Sin[(9*(c
+ d*x))/2] - 41895*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 41895*Log[1 - Cos[(c + d*x)
/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 3990*Sin[(11*(c + d*x))/2] - 11970*Sin[(13*(c + d*x))/2] + 5985*Log
[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 5985*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]
*Sin[7*(c + d*x)]))/(35840*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^7)

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Maple [A]  time = 1.213, size = 216, normalized size = 0.7 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{35840\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 5985\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{13/2}{a}^{5/2}-39900\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{11/2}{a}^{7/2}+5985\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{9} \left ( \sin \left ( dx+c \right ) \right ) ^{7}+98581\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{9/2}-95232\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{11/2}+1771\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{13/2}+39900\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{15/2}-5985\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{17/2} \right ){a}^{-{\frac{15}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x)

[Out]

-1/35840*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(15/2)*(5985*(-a*(sin(d*x+c)-1))^(13/2)*a^(5/2)-39900*(-a*
(sin(d*x+c)-1))^(11/2)*a^(7/2)+5985*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^9*sin(d*x+c)^7+98581*(-a*(sin
(d*x+c)-1))^(9/2)*a^(9/2)-95232*(-a*(sin(d*x+c)-1))^(7/2)*a^(11/2)+1771*(-a*(sin(d*x+c)-1))^(5/2)*a^(13/2)+399
00*(-a*(sin(d*x+c)-1))^(3/2)*a^(15/2)-5985*(-a*(sin(d*x+c)-1))^(1/2)*a^(17/2))/sin(d*x+c)^7/cos(d*x+c)/(a+a*si
n(d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{8}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^4*csc(d*x + c)^8, x)

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Fricas [B]  time = 1.33622, size = 1632, normalized size = 5.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/143360*(5985*(a*cos(d*x + c)^8 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2 - (a*cos(d*x +
c)^7 + a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^5 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^3 + 3*a*cos(d*x + c)^2 -
a*cos(d*x + c) - a)*sin(d*x + c) + a)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2
+ (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) +
(a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2
- 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(5985*a*cos(d*x + c)^7 + 1995*a*cos(d*x + c)^6 - 6811*a*cos(d*x +
c)^5 - 14633*a*cos(d*x + c)^4 - 5997*a*cos(d*x + c)^3 + 10097*a*cos(d*x + c)^2 + 1703*a*cos(d*x + c) - (5985*a
*cos(d*x + c)^6 + 3990*a*cos(d*x + c)^5 - 2821*a*cos(d*x + c)^4 + 11812*a*cos(d*x + c)^3 + 5815*a*cos(d*x + c)
^2 - 4282*a*cos(d*x + c) - 2579*a)*sin(d*x + c) - 2579*a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^8 - 4*d*co
s(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 - (d*cos(d*x + c)^7 + d*cos(d*x + c)^6 - 3*d*cos(d*x +
c)^5 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 + 3*d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**8*(a+a*sin(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out